Lagrangian Subvarieties of Hyperspherical Varieties Related to \(G_2\)

We consider two \(S\)-dual hyperspherical varieties of the group \(G_2\times\operatorname{SL}(2)\): an equivariant slice for \(G_2\) and the symplectic representation of \(G_2 \times \operatorname{SL}_2\) in the odd part of the basic classical Lie superalgebra \(\mathfrak{g}(3)\). For these varieties, we check the equality of the numbers of irreducible components of their Lagrangian subvarieties (zero levels of the moment maps of Borel subgroups’ actions), conjectured by M. Finkelberg, V. Ginzburg, and R. Travkin.